**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Modulo Indexing, Periodic Extension

The DFT sinusoids are all

periodichaving periods which divide . That is, for any integer . Since a length signal can be expressed as a linear combination of the DFT sinusoids in the time domain,

it follows that the “automatic” definition of beyond the range isperiodic extension, i.e., for every integer .Moreover, the DFT also repeats naturally every samples, since

because . (The DFT sinusoids behave identically as functions of and.) Accordingly, for purposes of DFT studies, we may defineallsignals in as being single periods from an infinitely long periodic signal with period samples:

Definition:For any signal , we define

for every integer .As a result of this convention, all indexing of signals and spectra

^{8.1}can be interpretedmodulo, and we may write to emphasize this. Formally, “” is defined as with chosen to give in the range .As an example, when indexing a spectrum , we have that which can be interpreted physically as saying that the sampling rate is the samefrequency as dc for discrete time signals. In the time domain, we have what is sometimes called the “periodic extension” of . This means that the input to the DFT is mathematically treated as

samples of a periodic signalwith period seconds ( samples). The corresponding assumption in the frequency domain is that the spectrum iszero between frequency samples.It is also possible to adopt the point of view that the time-domain signal consists of samples preceded and followed by

zeros. In this case, the spectrum isnonzerobetween spectral samples, and the spectrum between samples can be reconstructed by means ofbandlimited interpolation. This “time-limited” interpretation of the DFT input is considered in detail in Music 420 and is beyond the scope of Music 320 (except in the discussion of “zero padding interpolation” below).